Let \((W,S)\)  be an arbitrary Coxeter system and let \(H\)  be the associated Hecke algebra. In the landmark paper
[1]}, Kazhdan and Lusztig introduced the Kazhdan–Lusztig basis of \(H\)  and used it to partition the Coxeter
group \(W\)  into left, right and two-sided cells. Each two-sided cell is a union of left cells as well as a union of
right cells. These Kazhdan–Lusztig cells have connections with numerous objects from representation theory,
such as primitive ideals of universal enveloping algebras of Lie algebras [2]}, characters of
reductive groups over finite fields [3]}, and unipotent conjugacy classes of reductive groups
[4]}. Consequently, the determination of cells has been an important problem in representation
theory. The goal of this paper is to determine and count all left, right and two-sided cells of
\(\mathbf {a}\) -value 2 in \(\mathbf {a}(2)\) -finite Coxeter systems.
